Physics of Energetic Particles and Alfvén Waves*

Transcription

1 Athens, October 11 th 2017 Physics of Energetic Particles and Alfvén Waves* 1,2, Liu Chen 2, Zhiyong Qiu 2 and the NLED Team* 1 ENEA C.R. Frascati, Via E. Fermi 45 C.P. 65, Frascati, Italy 2 Institute for Fusion Theory and Simulation and Department of Physics, Zhejiang University, Hangzhou, P.R. China

2 Theory and simulation of energetic particle dynamics and ensuing collective behaviors in fusion plasmas [NonLinear Energetic particle Dynamics (NLED) ER15-ENEA-03*] NLED Wiki Page: NLED Project Team* Principal Investigator: Project Participants: Alessandro Biancalani, Alberto Bottino, Matthias Borchardt, Sergio Briguglio, Nakia Carlevaro, Claudio Di Troia, Remi Dumont, Matteo Faganello, Giuliana Fogaccia, Valeria Fusco, Xavier Garbet (till Jan.1st 2017), Ralf Kleiber, Axel Könies, Philipp Lauber, Zhixin Lu, Alexander Milovanov, Oleksiy Mishchenko, Giovanni Montani, Gregorio Vlad, Xin Wang, David Zarzoso External Collaborators (not supported by NLED funding): Ilija Chavdarovski (MPPC PostDoc Fellow), Michael Cole (PhD Student), Matteo Falessi (PhD Student till Feb. 1st 2017), Thomas Hayward (PhD Student), Malik Idouakass (PhD Student till Feb.1st 2017), Pierluigi Migliano (PostDoc Fellow), Ivan Novikau (PhD Student), Francesco Palermo (PostDoc Fellow), Mirjam Schneller (EUROfusion Fellow till Feb.1st 2016), Christoph Slaby (PhD Student), Xavier Garbet (CEA Faculty) Participating Research Institutions: Aix Marseille University, CEA Cadarache, ENEA Frascati, IPP Garching, IPP Greifswald

3 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 3 Outline (I) Introduction: Shear Alfvén Waves (SAW) and Energetic Particles (EP) (II) The General Fishbone Like Dispersion Relation: Unique theoretical framework for description of linear and nonlinear SAW dynamics (III) Nonlinear Physics: (III.A) Nonlinear Wave-Wave Interactions (III.B) Nonlinear Wave-Energetic Particle Interactions (IV) Energetic Particle Transport: Test Particle and self-consistent Energetic Particle Transport in the presence of multiple fluctuations (V) Conclusions and Discussion

4 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 4 (I) Introduction Hannes Alfvén [1942] discovered e&m waves can propagate in a conducting fluid (plasma) inthepresenceoffinite B 0 Magneto-Hydro-Dynamic Alfvén waves prevalent in nature & laboratory plasmas; e.g. Alfvén instabilities due to energetic particles in fusion plasmas Geomagnetic oscillations in solar-wind disturbed magnetosphere Significance of Alfvén waves: finite δe & δb energy and momentum exchange with charged particles Acceleration/heating/transport of charged particles: Solar corona heating, Transport across Earth s magnetosphere; and fast ion losses in burning fusion plasmas (ITER) Nonlinear coupling of fusion reactivity profiles with plasma stability and transport: long time-scale complex behavior mediated by energetic/alpha particles [C&Z Rev. Mod. Phys. 88, (2016); Springer Monograph in preparation]

5 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 5 Shear Alfvén continuous spectrum (1D) Shear Alfvén waves (SAW) are characterized by a continuous spectrum when the wave frequency varies continuously due to non-uniformity in the x- direction (B 0 = ẑb 0 (x)). They correspond to local (singular) fluctuations. ω 2 = ω 2 Al(x) k 2 lv 2 A(x). v g B 0,v 2 A(x) =B 2 0/4πϱ m We are interested in the ω Al t 1timeasymptoticbehaviors.Assuming, to be justified a posteriori, x k y, δj =0(vorticityEq.) becomes x [ ] 2 t + 2 ω2 Al(x) x δ ˆξ xl (x, t) =0. This equation can be straightforwardly integrated once and it yields x δ ˆξ xl (x, t) =Ĉ l (x)e ±iω Al(x)t,

6 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 6 Here, Ĉ l (x) isafunctiondependingonequilibriumnon-uniformities.now, note that, as ω Al t (the assumption x k y is justified), x = ±iω Al (x)t, (k x ) δ ˆξ Ĉ l (x) xl (x, t) = i Al(x)t. ω Al (x)te±iω From the condition δ ˆξ l =0,onederivesδ ˆξ yl =(i/k y ) x δ ˆξ xl ;i.e., δ ˆξ yl (x, t) = i k y Ĉ l (x)e ±iω Al(x)t. Insight in the dynamics associated with the resonant excitation of the SAW continuum and resonant wave absorption ( ω Al (x) [Chen74, Chen95]): the δ ˆξ xl component exhibits the characteristic (1/t) decayviaphase mixing of the continuous spectrum δ ˆξ yl shows undamped oscillations at local SAW frequencies

7 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 7 These properties are nicely demonstrated by the satellite observations of magnetic field fluctuations in the Earth s magnetosphere [Engebretson 87]. B R, B E and B N correspond to, respectively, δb x (δ ˆξ x ), δb y (δ ˆξ y ), δb.

8 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 8 The same behaviors have also been demonstrated by ideal MHD initial value numerical simulations of SAW dynamics in a cylindrical plasma [Vlad99]. Introducing (r, ϑ,z)ascoordinatesystemin a cylindrical plasma of periodic length 2πR 0 and radius a, δu(r, ϑ,z,t) t δξ = e i(nz/r0 mϑ) δû m,n (r, t).

9 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 9 The existence of local singular oscillations of the SAW continuous spectrum is further demonstrated by the Fourier frequency spectrum of δû r 1,1(r, t) at several radial locations, reflecting the spatial structure of the continuum frequency ω A (r/a)/ω A [Vlad99].

10 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 10 Kinetic theory I: thermal plasma response Break down of the ideal MHD assumption (k 2 ρ2 i 1) is suggested by k x ω A(x)t,whenapproachingtheSAWcontinuum. Proper treatment of Larmor radius scale SAW require kinetic theory analysis and yield the Kinetic Alfvén Wave (KAW) dispersion relation ω 2 = k 2 v 2 A(1 + k 2 ρ 2 K)=ω 2 A(1 + k 2 ρ 2 K), ρ 2 K ρ 2 i Most relevant new (w.r.t ideal MHD) dynamics on short scales are due to charge separation finite δe due to, e.g., FLR, finite electron inertia and plasma resistivity finite δe & wave-particle interactions Landau damping (col.less dissip.) singularities removed on short scales by finite radial energy propagation

11 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 11 Shear Alfvén waves in toroidal systems (2D) SAW continuous spectrum in one dimensional non-uniform plasma ω 2 = k 2 v 2 A = ω 2 A(ψ). Adapted from [Fasoli et al NAT16] Radial (ψ) singularmodestructures and resonant absorption (continuum damping) ω A(ψ) [Chen74, Chen95]. In higher dimensionality systems, such as nearly two-dimensional (2D) or three-dimensional (3D) toroidal devices, the main additional complication is due to the modulations of v A along B 0.

12 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 12 This causes the loss of translational symmetry for SAW traveling along B 0 and sampling regions of periodically varying v A. Similarly to electron wave packets traveling in a one-dimensional periodic lattice of period L, SAW in toroidal systems are characterized by gaps in their continuous spectrum, correspondingtotheformationofstanding waves at the Bragg reflection condition; i.e., 2L = lλ, λ 2π k, l N, { L =2πL0 =2πqR 0 connection length Considering that k k in this analogy, the Bragg reflection condition becomes k = l 2L 0, ω 2 = l2 v 2 A 4L 2 0, l N, modulation periodicity with v A representing now some typical value of the Alfvén speed on the reference magnetic surface.

13 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 13 Examples: l =1: ToroidalAlfvénEigenmode(TAE)gap[Pogutse78,D Ippolito 80, Kieras 82] l =2: EllipticityinducedAlfvénEigenmode(EAE)gap[Betti91] l =3: Non-circularityinducedAlfvénEigenmode(NAE) gap [Betti 91] Alow-frequency(l =0)gapalsoexistsbecauseoffiniteplasmacompressibility [Chu92, Turnbull 93] at ω β 1/2 i v A /R 0.Beta-inducedAE(BAE). Major breakthrough: nearly undamped fluctuations exist in frequency gaps near SAW acc. points (vanishing continuum damping ω A(ψ) =0) Theoretical prediction [Cheng, Chen, Chance 85] Radial (local) potential well due to geometry and equilibrium non-uniformity [C&Z POP14,RMP16] EP effects may be non-perturbative [Chen 84,94] (X.Wang,Z.Lu) Experimental observation [Heidbrink et al 91; Wong et al 91]

14 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 14 Alfvén Eigenmodes and Energetic Particles Modes Toroidal Alfvén Eigenmodes (TAEs) [Cheng, Chen Chance 1985] and Energetic Particle Modes (EPMs) [Chen 1994] observed in toroidal devices NSTX δb [Fredrickson et al. 2006] On left, bursting, chirping EPM-like modes (non-perturbative). Evolutions to nearly coherent, TAE-like modes on right.

15 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 15 Kinetic theory II: excitation by En. Particles Fishbone instability in PDX [McGuire; 1983] Suprathermal SAW fluctuations excited during perpendicular (to B) beaminjection experiments. Symmetry-breaking perturbations significant ( 30%) fraction loss of beam ions!!

16 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 16 Insights from PDX Fishbone observations: [White et al PF83] Excitations via wave-particle interactions tapping EP s finite P EP expansion free energy. Magnetically trapped charged particles precess in φ Precessional frequency ω d E = v 2 /2 ω = ω d resonant particles secularly move in the radial (R) direction

17 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 17 Gyrokinetic simulation of SAW (1) (NLED) AE spectrum in ITER [Gorelenkov 14, Lauber 15] (LIGKA [Lauber 05]). frequency [khz] NAE EAE TAE s!/"<0.05% 0.05%<!/"<0.5% 0.5%<!/"<2.5% 2.5%<!/"<50% electrostatic potential [arb. units] s ITER 15 MA scenario (LIGKA) [Lauber15] SAW continuum with local damping as color code Typical TAE mode structures Automatically obtained growth rates of various TAE branches using HAGIS-LIGKA code [T. Hayward-Schneider&Ph. Lauber 2017]

18 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 18 Gyrokinetic simulation of SAW (2) AE spectrum in W7-X [A. Könies et al 2015, 2017] (CKA-EUTERPE [A. Könies et al 2015]). ω/ω A SAW Continuous Spectrum m = 4 m = 5 m = 5 m = 6 TAE(m=6,7) mode m = 4 m = 7 Re(φ), normalized (NLED) (A. Mishchenko) AE mode structures m= 7 n=-6 m= 6 n=-6 m= 5 n=-6 m= 4 n=-1 m= 5 n=-1 fast-ion density norm. toroidal flux norm. toroidal flux In collisionless fusion plasmas, hybrid-kinetic and/or gyrokinetic descriptions are necessary and are becoming routine [C&Z RMP16].

19 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 19 (II) General Fishbone Like Dispersion Relation Kinetic Energy Principle General Fishbone Like Dispersion Relation (GFLDR) provides a unified theoretical framework [Z&C POP14] i s Λ(ω) =δŵf + δŵk(ω) s =magneticshear Λ(ω): inertia (kinetic energy) due to background plasma Structures of continuum, gaps,andresonantabsorption δŵf: δw (potentialenergy)due to background plasmas existence of discrete AEs (different types; depending on equilibrium) δŵk: δw (activepotentialenergy)duetoeps instability mechanisms & new unstable modes: EPMs [Chen 1994] Simple limit [Chen, White, Rosenbluth 1984]; [Coppi & Porcelli 1986] Λ(ω) =ω/ω A (ω A = v A /qr 0 ), δŵf 0 fishbone E ωd F EP δŵk F EP (r, E): EP distribution function ω d ω r v

20 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 20 The fishbone-like dispersion relation demonstrates the existence of two types of modes (note: Λ 2 = k 2 q2 R 2 0 is SAW continuum): adiscretegapmode,oralfvén Eigenmode (AE), forireλ 2 < 0; an Energetic Particle continuum Mode (EPM) for IReΛ 2 > 0. For AE Core plasma and non-resonant fast ion response provide a real frequency shift away from continuum accumulation point (Λ = 0); resonant wave-particle interaction gives the mode drive. δŵf +IReδŴk > 0: AEfrequencyis above the accumulation point δŵf +IReδŴk < 0: AEfrequencyis below the accumulation point For EPM iλ represents continuum damping [Chen et al 84, Chen 94] IReδŴ k (ω r )+δŵ f =0 determines ω r, (non perturbative) γ/ω r =( ω r ωr IReδŴk) 1 (IImδŴk s Λ) determines γ/ω r

21 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 21 Non-perturbative energetic particle response Comparing the GFLDR and structure of SAW continuum [Z&C POP14, C&Z RMP16, POP17]. (X.Wang,Z.Lu) GFLDR for AE/EPM SAW continuous spectrum i s Λ(ω) =δŵ f + δŵ k (ω) Λ 2 (ω) =k 2 (ψ)q 2 R 2 0 =(nq m) 2 Equilibrium geometry and non-uniformity of both core plasma (δŵf) and energetic particles (δŵ k (ω)) determine effective k of radially bound state potential well that determines mode frequency and mode structure. Recent interest for BAE and Alfvén Acoustic (BAAE [Gorelenkov et al 2007]) fluctuations: Re(δŴf + δŵk) < 0 Non-perturbative EP response in δŵ f > 0coreplasma[C&Z POP17] Confirmed in GK simulations [Liu et al NF17], [Bierwage and Lauber NF17] that also emphasize crucial role of kinetic core plasma response, consistent with theory [Chavdarovski 09,14].

22 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 22 Anti-hermitian EP response crucial for breaking the radial/parallel symmetry of mode structure [R. Ma et al POP15; Z. Lu et al 17]. Also important for momentum transport [Z. Lu et al 17]. RSAE ECE imaging in DIII-D [Tobias et al PRL11] Importance of k r = Rek r + iimk r [Lu et al 17] BAE mode structure (c r =60,c i =50,r 0 =0.5) exp [i(c r + ic i )(r r 0 ) 2 + inφ imθ] (Z. Lu)

23 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 23 (III) Nonlinear Physics Addresses SAW turbulence spectrum and spectral transfers Needed for proper assessments of heating/acceleration and transports Broad scope of ongoing research activities Formal solutions of nonlinear GFLDR: D = i s Λ L (δŵ f + δŵ k ) L [Zonca NF05, C&Z RMP16] D = i s Λ NL + (δŵf + δŵk) NL (III.A) Nonlinear Wave-Wave Interactions and spectral transfers: Λ NL. Understand and describe nonlinear processes in terms of breaking of the Alfvénic state cancellation of Reynolds and Maxwell stresses (III.B) Nonlinear Wave-EP Interactions and transports: δŵ NL k. Nonlinear dynamics of structures in the EP phase space phasespace zonal structures secular nonadiabatic resonant particle transport (τ TRANSP τ NL )onmeso-andmacro-scales

24 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 24 (III.A) Nonlinear Wave-Wave Interactions The pure Alfvénic state Nonlinear self-consistent SAW solution Infinite, uniform, ideal magnetohydrodynamic (MHD) fluid ϱ m ( t + u )u = P + J B/c u 0 =0=J 0, B 0 = B 0ˆb Nonlinear SAW Vorticity equation Quasi-neutrality: δj 0(Validforlow-frequencySAW) Ideal MHD constraint: δe 0 incompressible response c [ ] 2 (b 0 ) 2 V 2 A 2 t 2 δφ +4π t ( δj (2) )=0 δj (2) = (c/b 0)b 0 [Re + Mx] (Reynolds + Maxwell)

25 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 25 Pure Alfvénic state: Walén relation δu W /V A = ±δb W /B 0 [ t ± V A b 0 ]δφ W =0 Re + Mx =0 δj (2) =0 (wave-wave coupling suppressed) [ (b 0 ) 2 V 2 A 2 t ] δφw =0 δφ W :solutiontononlinearsawequations Nonlinear wave-wave interactions Breaking Alfvénic states: [C&Z POP13] Finite ion compressibility: ion sound perturbations along B Microscopic-scales (ρ i ) Kinetic Alfvén Waves Enhanced electron-ion decoupling Enhanced δe Geometries: continuousanddiscretesawspectra [e.g., ToroidalAlfvénEigenmodes(TAE)].

26 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 26 Finite Ion Compressibility: Parametric Decays in Uniform Plasma (A) Ideal MHD macro-scale theories [Sagdeev & Galeev 1969] Resonant decay ω k matching conditions Ω 0 =(ω 0, k 0 )=Ω S + Ω A Backscattering: Counter-propagatingSAWs

27 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 27 Parametric Dispersion Relation ϵ S ϵ A = C I eδφ 0 /T e 2 ϵ S : ISW ϵ A : SAW C I O(k ρ 2 2 i )cos 2 θ, Coupling maximizes around θ =0, π; k 0 k

28 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 28 (B) Gyrokinetic micro/meso-scale theory [C&Z, EPL 2011] k ρ i O(1) Parametric Dispersion Relation ϵ SK ϵ A K = C K eδφ 0 /T e 2 ϵ SK : Kinetic ISW ϵ A K : KAW [ (Ωi ) ] 2 C K O (k ρ i ) 6 sin 2 θ, ω 0 Maximizes around θ ±π/2 and k ρ i O(1) Simulation by Y. Lin et al. (PRL 2012)

29 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 29 (C) Quantitative & Qualitative differences [C&Z POP13] C I cos 2 θ k k 0 C K sin 2 θ k k 0 Qualitative implications to turbulence and transports Mode converted KAW k 0 k 0rˆr MHD regime: k k rˆr no P θ breaking little transport! Kinetic regime: k k θ ˆθ large P θ breaking significant transport! cross-scale couplings with excitation by EPs! C K /C I (Ω i /ω 0 ) 2 (k ρ i ) 4 For k ρ i 2 > ω 0 / Ω i O(10 2 ), kinetic effects are qualitatively and quantitatively crucial for SAW turbulence dynamics and associated transports.

30 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 30 Zonal Structures by Toroidal Alfvén Eigenmodes [C&Z, PRL, 2012] Zonal structures coherent micro/meso-scale radial corrugations of equilibrium in toroidal device plasmas. Examples: Zonal Flow Zonal Current [More generally: phase-space zonal structures] [Z&CNJP15;C&ZRMP16]

31 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 31 Zonal structures spontaneously excited by micro/meso-scale turbulence due to plasma instabilities. Zonal structures scatter instability turbulence to shorter-radial wavelength stable domain nonlinearly damp the instability. Self-regulation of plasma instabilities! In toroidal plasmas continuous and discrete spectra Continuous spectrum ω 2 = k 2(r)V A 2 (r) Re + Mx 0 negligible nonlinear contributions (Alfvénic State) Discrete spectrum AEs finite nonlinear contribution Spontaneous excitation of zonal structures via modulational instability of the radial envelope of a finite-amplitude TAE pump wave. Modulational stability condition Spontaneous excitation threshold: δb r /B 0 0 > O(10 4 ) Competitive nonlinear saturation process!

32 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 32 Other discrete AEs (e.g., BAE [Z.Qiu et al NF16]) can also break the Alfvénic State and stimulate interesting nonlinear wave-wave interactions. Nonlinear cross section of ZS generation by AEs is enhanced by fine radial structures w.r.t. DW turbulence case [Z.Qiu et al NF17]. strongly ballooning DWs weakly ballooning AEs EPs,viapressure-curvature coupling (δŵk NL ), can compete with core plasma (Reynolds/Maxwell stress) inzs formation [Z.Qiu et al POP16] Forced driven excitation of ZS![Z.Qiu et al POP16]: consistent with earlier sim. results [Y. Todo et al NF10; Z. Wang et al 17; A. Biancalani etal17]

33 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 33 NL GK simulations [Biancalani et al IAEA16] (ORB5 code) provide evidence of forced driven excitation of GAM (geodesic acoustic modes) by EP driven AEs, consistent with theory [Z.Qiu et al NF16]. EP driven GAM (EGAM) (anisotropic EPs) can play very important roles in cross-scale couplings with DW turbulence [D. Zarzoso et al 13 17], [R. Dumont et al 13], [J.-B. Girardo et al 14], [A. Biancalani et al 16].

34 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 34 (III.B) Nonlinear Wave-EP Interactions When considering transport processes due to nearly periodic fluctuations with γ/ω 1, wave-particle resonances play a crucial role both in wave excitations as well as transport processes [Chen JGR99]. Historically: nonlinear dynamics of 1D uniform Vlasov plasmas Phasespace holes (lack of density w.r.t. surrounding phase-space) and clumps (excess of density w.r.t. surrounding phase-space): extensively investigated after pioneering work by Bernstein, Greene and Kruskal (BGK) [Phys.Rev57]. Nonlinear dynamics of phase-space holes and clumps in the presence of sources and collisions: widely adopted by Berk, Breizman and coworkers (review by [Breizman PPCF11; Sharapov et al NF13]) 1D uniform beamplasma system as paradigm for nonlinear behaviors of Alfvén Eigenmodes near marginal stability [Berk PFB90].

35 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 35 EP-SAW interac- The beam-plasma system vs. tions in tokamaks Similarities can be drawn but strong differences and peculiarities emerge depending on drive strength [Zonca et al NF05; NJP15; C&Z RMP16]: Advantages of using a simple 1-D system for complex dynamics studies [Berk PFB90]; [Review by Breizman PPCF11] Roles of mode structures, non-uniformity and geometry in determining nonlinear behaviors [Zonca NF05, PPCF06]; [C&Z NF07, RMP16] for 1 γ/ ω > (depending on resonances)

36 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 36 Resonance detuning and radial decoupling Particle phase space diagnostics (Hamiltonian Mapping) [Briguglio et al POP14] (HMGC code [Briguglio et al POP95,98]) (X Wang) weak drive strong drive resonance detuning radial decoupling

37 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 37 Importance of geometry Effect of resonance detuning/radial decoupling on growth-rate scaling of saturation amplitude [Wang et al POP16] (HMGC code [Briguglio et al POP95,98]) (X Wang) Different behavior of co- and counterpassing particles due to resonance frequency with broader/narrower radial profile Geometry! Transition from quadratic γ-scaling (resonance detuning/wave-particle trapping) to linear γ-scaling (radial decoupling) consistent with simple theoretical model [Wang et al PRE12; POP16] Results confirmed by other NLED codes: EUTERPE [A. Könies et al 15]; [M. Cole et al 16].

38 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 38 Transport due to phase-space zonal structures Importance of wave-ep interactions Role of phase space zonal structures (PSZS) that are undamped by collisionless processes [Zonca et al NJP15]. Counterpart of zonal structures (ZS) (reflection of equilibrium corrugations/deviation from local equilibrium) PSZS regulate transport processes on transport time scale and longer [M. Falessi ArXiV 2017]. Evolution equation for renormalized particle distribution function (f 0 ) [Zonca et al NJP15]; [C&Z RMP16] Dyson Equation Nearly coherent NL interaction Importance of phase locking and phase bunching Break down of QL description (non-perturbative/adiabatic)

39 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 39 Application: Fishbone Nonlinear Theory [Zonca et al 2007; C&Z RMP16] based on the GFLDR: i s Λ(ω) =δŵf + δŵk (ω F 0EP ) Resonant EPs convect outward with radial speed δu n Nonlinear saturation occurs when δu n /γ L r s [r s mode structure width Wave-EP interaction domain] Consistent with numerical simulation results by [GY Fu et al POP 2006]. [Vlad et al., 2012] simulation results Near marginal stability regime explored by [M. Idouakass et al POP16; 2017 tbs] analytically and numerically Electron-fishbone simulation results with HMGC code [Vlad et al NF13]; [Vlad et al NJP16].

40 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 40 Non-perturbative NL EPM dynamics x , 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = X1 t=60 x , 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = X10 t= Zonca et al. IAEA, (2002) 8, 4 9, 4 10, 4 11, 4 12, 4 13, 4 14, 4 15, 4 16, 4 φ m,n (r) t/τ A0 = X30 t= δα H r/a δα H r/a δα H r/a NL distortion of free energy SR C r/a r/a r/a EPM convective amplification and EP secular transport Avalanches

41 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 41 (IV) Energetic Particle Transport Present understanding of nonlinear wave-wave and wave-particle interactions is reasonably sound and complete [C&Z RMP16]. One crucial open issue remains the realistic prediction of global transport of EPs and fusion products and their impact on the system material walls. Collective oscillations excited by EPs in burning plasmas are characterized by a dense spectrum of modes with characteristic frequencies and spatial locations [C&Z NF07, RMP16]. Standard approach usually relies on: Near marginal stability Ansatz Quasilinear description Test-particle studies Advanced NL GK analyses Reduced models (advanced) Widely & successfully adopted Fundamental issues remain

42 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 42 Advanced NL GK analyses: The natural framework for predicting EP transport by Alfvénic and MHD fluctuations NL GK models [many groups/including NLED Team]: easy coupling with DWT; buttechnicalissuesremain Hybrid MHD-Gyrokinetic models [Briguglio, Vlad et al]; [Todo et al]; [Gorelenkov, Fu et al]: kinetic physics for low frequency may be an issue/coupling with DWT? Gyrofluid models [Spong et al.; Staebler et al.]: nonlinear closure remains an issue Reduced descriptions are needed (beyond QL/Test-particle) Outstanding theoretical issue: transport on long time scales (longer than characteristic transport time) [M. Falessi et al.] transport of phase space structures (PSZS) and deviation from thermodynamic equilibrium (meso-scales) interplay of collisional and fluctuation induced transport

43 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 43 Nonlinear energetic particle transport in the presence of multiple Alfvénic waves in ITER: reduced GK analysis with HAGIS-LIGKA [M. Schneller et al ] Linear GK mode structures (LIGKA code [Lauber 05]) with nonperturbative EP Amplitude evolution consistent with EP transport Evidence of break-down of QL description Importance of modes of the linear stable and unstable spectrum. Confirmed in recent assessment [T. Hayward-Schneider, 2017]. Saturation level can be different (higher) than single-mode saturation cross-scale coupling mediated by EP [C&Z RMP16]. Confirmed in Hybrid MHD-GK simulations (HMGC) [Vlad et al 2017]. [M. Schneller et al PPCF15]

44 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 44 Correspondence of SAW and BoT problems: implications to EP transport (reduced model) [N. Carlevaro et al JPP15; ENT16] Correspondence/mapping r v must be established preserving nonlinear displacement (not growth rate) [N. Carlevaro et al 2017] Importance of linear unstable as well as stable spectra. Crucial role of cross scale couplings mediated by EP: modesmaysaturateathigher value than single mode saturation. Meso-scale (spatiotemporal) behavior is due to self-consistent evolution of fluctuation intensity on the same time scale of particle transport. Importance of phase bunching and phase locking beyond the QL paradigm.

45 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 45 (V) Conclusions and Discussion (V.1) SAW in plasmas confined by realistic B rich, interesting physics (V.2) Nonuniformities and geometries Continuous spectrum, singular resonant absorption, mode conversion to KAW, frequency gaps, discrete Alfvén eigenmodes. (V.3) Nonlinear physics: Nonlinear wave-wave interactions: Compressibility, geometries, microscopic kinetic effects Breaking the Alfvénic states Qualitative and quantitative effects on parametric decay instabilities, generation of nonlinear equilibria, excitation of convective cells, and zonal structures

46 17th European Fusion Theory Conference (EFTC), Athens Oct (2017) 46 Nonlinear wave-ep interactions: Frequency chirping and phase locking mode particle pumping EP radial redistribution wave-particle decoupling due to finite mode widths. Self-consistent interplay of NL mode dynamics and EP transport EPM-EP Avalanches; Fishbones; Secular transport (V.4) Realistic plasma nonuniformities, B geometries, mode structures Play crucial roles in the linear and nonlinear SAW and non-perturbative EP dynamics!! Need to go beyond Quasi-Linear and/or Test-particle EP transport (V.5) Careful in-depth analyses and cross-checks among experiments/observation, theory, andrealisticfirst-principlesimulations advances in Alfvén wave and EP physics Intellectually exciting! Practically important!